$\begingroup$I don't have the math on hand to address this, but it would be interesting to use Earth as the test case: can something be put in orbit around the moon? The answer is almost certainly yes.$\endgroup$
– Isaac KotlickyMar 29 '15 at 4:29

$\begingroup$If you are goingnto write hard SF, you ought to learn a bit about physics.$\endgroup$
– JDługoszMar 29 '15 at 16:48

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$\begingroup$@IsaacKotlicky: The question is not so much "can anything be put in the orbit" but "How long will it stay there?" - in case of human-made satellites, several years at most, despite lack of air friction.$\endgroup$
– SF.Apr 17 '15 at 15:16

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$\begingroup$Can it be a satellite whose orbit is artificially/magically adjusted? I'm guessing, 'no', but you might want to make the answer, 'yes'.$\endgroup$
– MikeyApr 17 '15 at 20:29

7 Answers
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Sattelites of the moon are unstable: too close and it notices the lumpiness and does not make a nice ellipse, so after a while will crash. If too far, it will fall out and orbit the Earth instead.

Look at the “Hill Sphere” represented as a topological map, and you get the feeling for how the gravity of the Earth interferes, leaving only a small region where orbits look like they would go around in a normal manner.

But you can’t orbit that close, since the moon is not just lumpy but off ballance. As it turns out, certain high-inclination orbits work out better, on paper anyway.

So first, you need to find out what makes the hill sphere bigger. I don’t know off hand, but I suppose the moon can be larger and farther away. But if the moon is far from the primary, it has the same issue and will end up orbiting the sun, or easily peturbed from other bodies in the solar system.

Also, our moon is rather lopsided. A more suitable situation would be to have the moon be especially symmetric. Perhaps it could be liquid? That would allow the sattelite to orbit close in and remain stable.

Throw in some other reason to remain stable, like the all-powerful resonance. Make it slightly eccentric, make the sub-sattelite off-ballance and in an odd half multiple of its period, which is also a small multiple of the moon’s period, and furthermore have another large moon with a period an exact integral multiple of the moon’s.

If that’s not quite right, it’s at least believable to the audience, if they don’t solve the math for real.

Then I read the Keck proposal for parking an asteroid in lunar orbit. I was surprised when they mentioned the possibility of parking the rock in a stable lunar orbit that would need no station keeping.

One example of a body in a DRO is Jupiter's Moon S/2003 J2. This moon's semi-major axis is nearly 30 million kilometers. Jupiter's Hill Sphere has a radius of of 52 million kilometers. The S/2003 J2 Wikipedia article says

The limits of Jupiter's gravitational influence are defined by its
Hill sphere, whose radius is 52 gigametres (0.35 AU). Retrograde moons
with axes up to 67% of the Hill radius are believed to be stable.
Consequently, it is possible that even more distant moons of Jupiter
may be discovered.

Our Moon's Hill Sphere has a radius of about 60,000 kilometers. In my orbital sims I have placed objects in a retrograde lunar orbit having semi-major axis 50,000 km. These orbits have lasted centuries. I would imagine a DRO at 40,000 kilometers could last eons. And this is high enough that lunar mascons wouldn't destabilize the orbit.

In a science fiction setting, a moon with a larger Hill Sphere is plausible. DROs are long term stable in the outer regions of the Hill Sphere.

$\begingroup$That's the first I heard of it, too. I recall certain high-inclination orbits did better than typical but that was all. Retrograde: hmm, let me guess: it won't fall off into an independent Earth orbit because the Moon's speed (already the correct orbital speed for that radius) and the asteroid's are cancelling. Angular-momentum is wrong. But I can't visualize it.$\endgroup$
– JDługoszApr 15 '15 at 5:34

$\begingroup$@JDługosz indeed. I can see how a near 60,000 km prograde orbit would fall towards earth as it nears EML1 or how it'd fly away from both earth and moon as it nears EML2 -- it's orbital velocity is added to the moon's at EML2 and subtracted at EML1. A retrograde is just the opposite and thus more robust.$\endgroup$
– HopDavidApr 16 '15 at 19:34

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$\begingroup$This has messed up some of my models, though. I used to erroneously assume orbits were time reversible.I was surprised to learn retrograde lunar orbits act differently than prograde.$\endgroup$
– HopDavidApr 16 '15 at 19:34

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$\begingroup$Right, you can't just reverse some of the parts. The common contour map analogy of a hill volume doesn't have enough dimensions if you have more than 2 bodies, I wonder how you can present the volume with sensitivity to orbital parameters of the object moving within it. It commonly shows a value meaning some closed orbit passes through this point.$\endgroup$
– JDługoszApr 16 '15 at 22:17

Yes, they can. However, generally satellites of moons will get pulled out of orbit.

NASA has orbited spaceships and lunar modules around moons, proving that it can be done. However, the tidal forces of the planet will soon pull the moon's satellite out of orbit, probably into orbit around the planet. Maybe a large moon far away from its planet could support a satellite, since its own gravitational field would be stronger than the field of the planet at that distance.

If the Moon's moon is very small, you can easily have one. But the problem is it wouldn't be visible from Earth.

If you want the Moon's satellite be visible from Earth with the naked eye, it has to be some decent fraction of the size of the Moon. But now you will have a 3-body system. This is not good.

This is because the Moon's moon is large enough that its gravity can't be ignored. Usually, 3-body systems are unstable. So you would either have a collision or one of the bodies (probably the smallest will be flung out)

The solution to your problem might be placing the smaller moon in one of Lagrangian points. Maybe L4.

Having said that, it still may not work, as placing a large object in L4 might destabilize the system.

Another solution maybe placing the smaller moon in close orbit around Earth.

$\begingroup$Could you explain what lagrangian point is?$\endgroup$
– DonyorMMar 30 '15 at 14:16

$\begingroup$Lagrange points also require that the third body have negligible mass relative tot he other two.$\endgroup$
– smithkmMar 30 '15 at 19:57

$\begingroup$Too late for April Fools? Too bad$\endgroup$
– JDługoszApr 16 '15 at 22:21

$\begingroup$+1 for being the only answer that I noticed actually addressed the topic of the question: "Can it have a *visible* satellite?" Whether it can have a stable satellite at all is an interesting conversation, but the subject was as to whether a satellite would be visible.$\endgroup$
– LoduwijkJan 19 '18 at 16:56

moon is a satellite of earth, earth is a satellite from sun, so 2nd level satellites allready exist, but this is possible because sun/earth and earth/moon are very different and moon is not much perturbated by sun.

$\begingroup$Welcome to the world-building stack exchange, DJousto! We like to have our answers be much more detailed and have references where we can have them. This is especially true of such physics-based questions such as this one. Technically, this does not directly answer the question, because you sidestep the real question here: can a moon have their own moon? If you address the question directly and very well, you'll get votes and more rep!$\endgroup$
– PipperChipMar 30 '15 at 6:48

$\begingroup$Hmm, seems to me that he makes a good point. We know that a 3-level system, like Sun-Earth-Moon, is not only possible, but exists and is to all appearances stable. We know that a 4-level system, like Sun-Earth-Moon-Apollo Command Module, is possible and stable for at least a few days. We know this by actual experience and observation, which beats theoretical calculations. So the only question is, how big could the 4th-level moon be and the system still remain stable? Is there any fundamental difference between a 4-level system and a 2 or 3 level system? Probably not.$\endgroup$
– JayMar 30 '15 at 19:18

I just thought of something when I read the question again. Assuming that being physically small and low-mass compared to its primary makes an improvement, it could be small but still visible for reasons other than reflected sunlight on common rock.

From the perspective of a lifeform, the motion of the planets is unchanging, but in reality it is always shifting and eventually unstable. So you could permit a situation that was not stable given some plausible story about how it came to be - most systems are accretion-based, so the orbit has to be stable enough for the orbiting debris to accrete and become a moon.

As an example, the currently accepted theory is that a smaller proto-Earth collided with a Mars-sized body, leaving a larger planet (the current Earth) and a vast debris field from the second body which has accreted into our Moon. The Moon is slowly moving away from Earth as the Earth's spin slows, so this is not permanently stable either.

A possible story could be that a stable Planet+Moon pair meets an impactor flung out by a complex path around the giants in the system, which collides with the planet as above. If the original Moon was in an inclined orbit compared to the impactor's path, and suffered a glancing blow, there could be a period of time in which both moons orbited each other and the planet. Albeit this would not be long-lived, and would involve a lot of debris raining on the planet for a while.

Fun products of this regarding stories would be:

A realisation of just how precarious the motion of the heavens is, perhaps explaining some environmental changes - even an early astronomer might be able to calculate some ongoing drift of their orbit and infer that they were due to collide

An icy second moon could be warmed by the tidal forces, permitting an ecosystem submerged beneath an icy crust to come to the surface

The Hill Sphere of a moon is very small, so for this to work for any length of time the two bodies would have to be rather close to each other

The oddness of the situation could be attributable to a meddling alien race either intentionally setting it all up or unintentionally creating the setup during other activities. An experiment to see if a race advances faster with a very clear example of the laws of gravities above them?